On the stochastic engine of transmittable diseases in exponentially growing populations

The paper’s Theorem 3 states and proves global asymptotic stability of the endemic equilibrium E* = (x1, F1(x1), 0) for the two-strain system (7) under x1 < min(x2, 1), using a custom integral-type Lyapunov function and LaSalle’s invariance principle . The candidate solution proves the same claim with a different, simpler logarithmic Lyapunov function, identifies the largest invariant subset of {Ẇ = 0}, and carefully clarifies boundary effects (y = 0) that the paper only implicitly assumes away. Both arguments are correct; the paper’s proof is terse but sound, relying on Theorem 1’s compact positively invariant set and a monotonicity property of F1(x) . The candidate’s proof supplies a clean alternative Lyapunov function and an explicit invariance check on the zero-derivative set. Minor wording in the paper (“positive octant”) could mislead about including the invariant face y = 0, but the intended domain is the interior (y > 0), consistent with the Lyapunov domain of definition.